Euclidean Lattices: Algorithms and Cryptography
نویسنده
چکیده
Euclidean lattices are a rich algebraic object that occurs in a wide variety of contextsin mathematics and in computer science. The present thesis considers several algorithmicaspects of lattices. The concept of lattice basis reduction is thoroughly investigated: in par-ticular, we cover the full range of time-quality trade-offs of reduction algorithms. On thefirst hand, we describe and analyse fast algorithms for finding a relatively short basis (LLL-reduced basis) of an arbitrary given lattice. On the second hand, we propose novel analy-ses for (slower) algorithms that compute very short bases (HKZ-reduced and BKZ-reducedbases). This study on how to efficiently solve algorithmic problems on lattices is completedby a constructive application exploiting their apparent hardness. We propose and analyzecryptographic schemes, including the NTRU encryption function, and prove them at leastas secure as well-specified worst-case problems on lattices.
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